This paper presents a medical robotic system for deep orbital interventions, with a focus on Optic Nerve Sheath Fenestration (ONSF). ONSF is a currently invasive ophthalmic surgical approach that can reduce potentially blinding elevated hydrostatic intracranial pressure on the optic disc via an incision on the optic nerve. The prototype is a multi-arm system capable of dexterous manipulation and visualization of the optic nerve area, allowing for a minimally invasive approach. Each arm is an independently controlled concentric tube robot collimated by a bespoke guide that is secured on the eye sclera via sutures. In this paper, we consider the robot’s end-effector design in order to reach/navigate the optic nerve according to the clinical requirements of ONSF. A prototype of the robot was engineered, and its ability to penetrate the optic nerve was analysed by conducting ex vivo experiments on porcine optic nerves and comparing their stiffness to human ones. The robot was successfully deployed in a custom-made realistic eye phantom. Our simulation studies and experimental results demonstrate that the robot can successfully navigate to the operation site and carry out the intervention.